There are many open problems which are fairly easy to state, also one might need some basic definitions, for example derived functors. I will provide mostly pointers to the ones I know, you can google for more details (one can easily fill many projects with each topic below). Interestingly, most people do not consider the Monomial conjecture as a problem about Artinian rings (the word Artinian only appear as in the definition of a system of parameters).
Representation theory of Artin algebras: you can start with the list at page 411 of this book.
Numerics of Betti numbers, Hilbert functions: perhaps the most famous one is the Buchsbaum-Eisenbud-Horrocks Conjecture;
If $R$ is a regular local ring of dimension $n$ and $M$ is an Artinian $R$-module, then the rank of the $i$-th module in the minimal free resolution of $M$ is at least $n \choose i$. (weaker version, but as open: the sum of the ranks is at least $2^n$, graded version is also open).
An important open problem is to characterize the Hilbert functions of Gorenstein graded artinian algebras. See the surveys by Irena Peeva or Valla for some other questions.
Homological problems: Here the big one, and also one of the simplest looking, is the Auslander-Reiten Conjecture
If $R$ is commutative Artinian ring and $M$ is a finitely generated $R$-module such that $\text{Ext}^i(M, M\oplus R) = 0$ for all $i>0$, then $M$ is projective.